Mathematical logic and foundations Definition
Transitive relation
Builds on Binary relation
Definition
Let be a binary relation on a set . The relation is transitive when, for all , and imply .
Remarks
The relations , and on are transitive, and so is divisibility on . "Is a parent of" is not transitive. The smallest transitive relation that contains is the transitive closure of .
Reading path
To understand this page from the ground up, read these 11 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Element definition
- Proposition concept
- Set definition
- Implication definition
- Predicate definition
- Ordered pair definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- Binary relation definition
- Transitive relation definition
Where this leads
193 pages build on this one.
- Hasse diagram
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Partial order
- Equivalence classes form a partition
- Well-founded relation
- Total order
- Torus
- Möbius strip
- Zorn's lemma
- Groups of prime order are cyclic
- Correspondence between equivalence relations and partitions
- The reals have the cardinality of the continuum
- Equivalence class
- Order of an element divides the order of the group
- Fixed points of a p-group action
- Second Sylow theorem
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- Well-order
- Universal property of the quotient topology
- Lexicographic order
- The alternating group A5 is simple
- Antichain
- Chain
- Upper bound
- Abelian simple groups have prime order
- Cauchy's theorem
- Hausdorff maximal principle
- The number of Sylow subgroups divides the index
- Minimal element
- Continuum hypothesis
- Well-ordering theorem
- Converse of Lagrange's theorem fails for the alternating group A4
- Transitive closure
- Third Sylow theorem
- Equivalence relation induced by a function
- Cardinal number
- Frattini argument
- Ordinal number
- Preorder
- Size of an orbit times the size of the stabiliser
- Path component
- Equivalence relation
- Least element
- Maximal element
- Aleph-null
- Quotient topology
- First Sylow theorem
- Cardinality of the continuum
- Monotone function
- Circle as a quotient of the real line
- Order isomorphism
- Groups of order pq
- Burnside's lemma
- Greatest element
- g to the order of the group is the identity
- Transfinite induction
- Quotient set
- Relative CW complex
- Closure operator
- Translation surface
- Arnold tongue
- Linearly ordered group
- Weak ordering
- Lebesgue's theorem on differentiability of monotone functions
- Order type
- Ordinal arithmetic
- Discrete series representation
- Kleene algebra
- Classification of surfaces
- Lagrange's theorem
- Nilradical is the intersection of the prime ideals
- Oscillator representation
- Ultrafilter lemma
- Von Neumann universe
- Kolmogorov–Arnold–Moser theorem
- Galois group of a prime degree polynomial with two nonreal roots
- Fodor's lemma
- Ordinal analysis
- Woodin cardinal
- Naive set theory
- Stationary set
- Cantor normal form
- Veblen function
- Axiom of regularity
- Riesz space
- Kunen's inconsistency theorem
- Lp space
- Nichols–Zoeller theorem
- Singular cardinals hypothesis
- Standard map
- Hall's theorem on solvable groups
- Bourbaki–Witt theorem
- Partially ordered ring
- Ordered group
- Cantor's isomorphism theorem
- Baire function
- Order dimension
- Order embedding
- Proof that the order of an element divides the order of the group
- Interior operator
- Duality in order theory
- Arnold's cat map
- Upper set
- Proof-theoretic ordinal
- Ordinal analysis
- Proof-theoretic ordinal
- Archimedean equivalence classes
- Strict total order
- Lebesgue–Stieltjes integration
- Hölder's theorem on Archimedean groups
- Category of posets
- Birkhoff's subdirect representation theorem
- Suslin's problem
- Karamata's Tauberian theorem
- Integral operator
- Hall subgroup
- Characterizations of semisimple modules
- Covering relation
- Cohen's independence of CH
- f-ring
- Sylow theorems
- Hahn embedding theorem
- Frattini subgroup
- Lattice-ordered group
- Proof that groups of prime order are cyclic
- Action-angle coordinates
- Club set
- Sinai billiard
- Diamond principle
- Dense order
- Ordered ring
- Milnor's isospectral flat tori
- Set theory
- Jordan decomposition of a function of bounded variation
- Orbifold
- Proper forcing axiom
- Aronszajn tree
- Wiener–Ikehara theorem
- Epsilon nought
- Galois connection
- Torus knot
- Suslin line
- Orientation of a manifold
- Baer's criterion
- Real projective plane
- Idempotent semiring
- Alexander subbase theorem
- Cantor function
- Cofinality
- Tauberian theorems for integral transforms
- Transcendence degree
- Isotonic regression
- Liouville–Arnold theorem
- Normal function
- Gödel's consistency of CH and AC
- Left-orderable group
- Gentzen's consistency proof
- Ordered semigroup
- Cardinal characteristics of the continuum
- Erdős–Rado theorem
- Maximum and minimum
- Ultrafilter
- Boolean prime ideal theorem
- Strict partial order
- Bounded poset
- Residuated mapping
- Szpilrajn extension theorem
- Aleph number
- Down-closure
- Hahn series
- Von Neumann–Bernays–Gödel set theory
- Proof of Lagrange's theorem
- Class
- Existence of transcendence bases
- Positive cone
- Easton's theorem
- Maximal chain
- Pseudo-Anosov map
- Ordered vector space
- Fubini's theorem on differentiation
- Geometric realization
- Galois insertion
- Torus as a group
- Vaught conjecture
- Bunimovich stadium
- Maximal torus
- Symmetric and self-reciprocal kernels
- Stone–von Neumann theorem
- Martin's axiom
- Beth number
- Regular cardinal
- Orderable group
Further reading
- Web https://en.wikipedia.org/wiki/Transitive_relation, lead section
About this page
| Tag | 0CYRECG2YQ |
|---|---|
| URI | m2://0CYRECG2YQ |
| Kind | definition |
| Writing | W1 stated |
| MSC | 03E20 |
| Defines | Transitive relation |
| Uses | Binary relation |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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